At the heart of growing demands for nanotechnology is the need of ultrafast transistors whose response time is one of the key performance indicators. The response of a general quantum open system can be probed by sending a step-like pulse across the system and monitoring its transient current over time, thus making transient dynamics a very important consideration. Many experimental data show that most molecular device characteristics are closely related to the material and chemical details of the device structure. Therefore, first principles analysis, which involves quantitative and predictive analysis of device characteristics, especially their dynamic properties, without relying on any phenomenological parameter, becomes a central interest in the design of nanoelectronics.
The theoretical study of transient current dates back twenty years to when the exact solution in the wideband limit (“WBL”) was obtained by Wingreen et al. See, N. S. Wingreen et al., Phys. Rev. B 48, 8487(R) (1993), which is incorporated herein by reference in its entirety. Since then transient current has been studied extensively using various methods, including the scattering wave function, non-equilibrium Green's function (“NEGF”) approach, and density matrix method. See, J. Wang, J. Comput. Electron. 12, 343-355 (2013); S. Kurth et al., Phys. Rev. B 72, 035308 (2005)(hereinafter “Kurth”); G. Stefanucci et al, Phys. Rev. B 77, 075339 (2008)(hereinafter “Stefanucci”); Y. Zhu et al., Phys. Rev. B 71, 075317 (2005); J. Maciejko et al., Phys. Rev. B 74, 085324 (2006)(hereinafter “Maciejko”); R. Tuovinen et al., Phys. Rev. B 89, 085131 (2014)(herein after “Tuovinen”); R. Seoane Souto et al., Phys. Rev. B 92, 125435 (2015) and X. Zheng et al., Phys. Rev. B 75, 195127 (2007), all of which are incorporated herein by reference. The major obstacle to the theoretical investigation of transient current is its computational complexity. Many attempts have been made trying to speed up the calculation See Kurth and Tuovinen, as well as L. Zhang et al., Phys. Rev. B 86, 155438 (2012); B. Gaury et al., Phys. Rep. 534, 1-37 (2014); M. Ridley et al., J. Phys.: Conf. Ser. 696, 012017 (2016)(hereinafter “Ridley”); H A. Croy et al., Phys. Rev. B 80, 245311 (2009) and F J. Weston et al., Phys. Rev. B 93, 134506 (2016), all of which are incorporated herein by reference in their entirety.
Despite these efforts, the best algorithm for calculating the transient current from first principles going beyond WBL limit scales like TN3 using a complex absorbing potential (“CAP”), where T and N are the number of time steps and the size of the system, respectively. See, L. Zhang et al, Phys. Rev. B 87, 205401 (2013) (hereinafter “Zhang”), which is incorporated herein by reference in its entirety. It should be noted that if WBL is used, the scaling is reduced. See, the Ridley article. However, to capture the feature of the band structure of the lead and the interaction between the lead and the scattering region, WBL is not a good approximation in the first principles calculation. As a result, most of the first principles investigations of transient dynamics were limited to small and simple one-dimensional systems.
There are a number of problems, such as with magnetic tunneling junctions (MTJ) and ferroelectric tunneling junctions, where the system is two dimensional or even three dimensional in nature. See, Z. Y. Ning et al., Phys. Rev. Lett. 100, 056803 (2008) and J. D. Burton et al., Phys. Rev. Lett. 106, 157203 (2011), which are incorporated herein by reference in their entirety. For these systems, a large number of k points Nk have to be sampled in the first Brillouin to capture accurately the band structure of the system. For MTJ structures like Fe—MgO—Fe, at least Nk=104 k points must be used to give a converged transmission coefficient. See, D. Waldron et al., Phys. Rev. Lett. 97, 226802 (2006), which is incorporated herein by reference in its entirety. This makes the time consuming transient calculation Nk times longer, which is an almost impossible task even with high performance supercomputers. Clearly it is urgent to develop better algorithms to reduce the computational complexity.